All subjects · Mathematics · WME03 Mechanics 3

Angular speed and radial acceleration

Edexcel IAL Mathematics · WME03 Mechanics 3

A particle moving in a circle at constant speed is still accelerating, because its direction keeps changing. The acceleration points to the centre of the circle, and some force must provide it.

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Angular speed

Angular speed ω is the angle turned per second, in radians per second. One revolution is 2π radians, so n revolutions per minute is ω = 2πn/60 rad/s. The time for one revolution is 2π/ω. A particle at radius r has speed v = rω.

Radial acceleration

For motion in a circle of radius r at constant speed, the acceleration is directed towards the centre and has magnitude

rω2 = v2/r.

Pearson requires both forms. By Newton's second law the resultant force towards the centre is mrω2 = mv2/r.

Where the force comes from

There is no extra "centripetal force": it is the resultant of the real forces, such as a tension, friction or a normal reaction. For a particle on a rotating rough disc, friction provides it: mrω2 = F ≤ μmg, so the particle stays in place while ω2 ≤ μg/r.

Worked example

A coin is placed 0.15 m from the centre of a horizontal disc. The coefficient of friction is 0.4. Find the greatest angular speed of the disc for which the coin does not slip.

  1. Vertically R = mg. Friction provides the radial force: m(0.15)ω2 = F.
  2. No slipping: F ≤ 0.4mg, so 0.15ω2 ≤ 0.4 × 9.8.
  3. ω2 ≤ 26.13.

Answer: ω = 5.11 rad/s

Key idea: v = rω; acceleration towards the centre rω2 = v2/r; the resultant of the real forces supplies m rω2.

Check yourself

  1. Convert 45 revolutions per minute to radians per second, to 3 significant figures.
    Show answer

    4.71 rad/s. 45 × 2π/60 = 1.5π = 4.71 rad/s.

  2. A particle moves in a circle of radius 0.3 m with angular speed 4 rad/s. Find its speed.
    Show answer

    1.2 m/s. v = rω = 1.2 m/s.

  3. A particle moves in a circle of radius 1.5 m at a constant speed of 6 m/s. Find the magnitude of its acceleration.
    Show answer

    24 m/s2. v2/r = 36/1.5 = 24 m/s2.

  4. A particle of mass 0.2 kg moves in a horizontal circle of radius 0.5 m with angular speed 3 rad/s. Find the resultant force on it.
    Show answer

    0.9 N. mrω2 = 0.2 × 0.5 × 9 = 0.9 N, towards the centre.

  5. In the worked example, find the greatest angular speed, to 3 significant figures.
    Show answer

    5.11 rad/s. ω2 = 0.4 × 9.8/0.15, ω = 5.11 rad/s.

  6. A particle moves in a circle with angular speed 2.5 rad/s. Find the time for one revolution, to 3 significant figures.
    Show answer

    2.51 s. 2π/2.5 = 2.51 s.

  7. A particle moves in a horizontal circle at constant speed. In which direction is its acceleration?
    • Towards the centre of the circle
    • Along the tangent, in the direction of motion
    • Away from the centre
    • It has no acceleration
    Show answer

    Towards the centre of the circle. The velocity changes direction towards the centre all the time.

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Last updated 8 November 2026. Spotted a mistake? Tell us.